Hindawi Publishing Corporation
Boundary Value Problems
Volume 2010, Article ID 317369, 14 pages
doi:10.1155/2010/317369
Research Article
Comparison between the Variational Iteration
Method and the Homotopy Perturbation Method
for the Sturm-Liouville Differential Equation
A. Neamaty1 and R. Darzi2
1
2
Department of Mathematics, University of Mazandaran, Babolsar 47416-95447, Iran
Department of Mathematics, Islamic Azad University Neka Branch, Neka 48411-86114, Iran
Correspondence should be addressed to R. Darzi, r.darzi@umz.ac.ir
Received 28 October 2009; Accepted 10 April 2010
Academic Editor: Claudianor Alves
Copyright q 2010 A. Neamaty and R. Darzi. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We applied the variational iteration method and the homotopy perturbation method to solve
Sturm-Liouville eigenvalue and boundary value problems. The main advantage of these methods
is the flexibility to give approximate and exact solutions to both linear and nonlinear problems
without linearization or discretization. The results show that both methods are simple and
effective.
1. Introduction
The variational iteration method VIM 1–4 and homotopy perturbation method HPM 5–
8, proposed by He, are powerful analytical methods for various kinds of linear and nonlinear
problems. For example, the variational iteration method has been applied to autonomous
ordinary differential equation 9 and delay differential equation 10. Abdou and Soliman
applied this method to Schrodinger-KDV, generalized KDV, and Shallow water equations
11, Burger’s equations, and coupled Burger’s equations 12. Furthermore, Momani
and Abuasad 13 used VIM for Helmoltz partial equation. Also homotopy perturbation
method was successfully applied to Voltra’s integrodifferential equation 14, boundary value
problem 8, nonlinear wave equations 15, and so forth; see 16–20. In this paper, we exert
these methods for linear Sturm-Liouville eigenvalue and boundary value problems BVPs.
A linear Sturm-Liouville operator has the form
yt : Kyt λrtyt gt,
1.1
2
Boundary Value Problems
where
K−
d
d
pt
qt,
dt
dt
t ∈ I a, b,
1.2
and gt is known analytic function representing the nonhomogeneous term. Associated
with the differential equation 1.1 are the following separated homogeneous boundary
conditions:
α1 ya β1 y a 0,
α2 yb β2 y b 0,
1.3
where α1 , α2 , β1 , and β2 are arbitrary constants. For simplicity, we will assume that
pt, p t, qt, and rt are continuous. The values of λ for which BVP has a nontrivial
solution are called eigenvalues of L, and a nontrivial solution corresponding to an eigenvalue
is called an eigenfunction.
The paper is organized as follows: in Sections 2 and 3, an analysis of the variational
iteration and homotopy perturbation methods will be given. In Section 4, we apply HPM to
solve Sturm-Liouville problems. We present 3 examples to show the efficiency and simplicity
of the proposed methods in Section 5. Finally, we give our conclusions in Section 6.
2. He’s Variational Iteration Method
To illustrate the basic concept of He’s variational iteration method 1–4, we consider the
following nonlinear differential equation:
Lu Nu gt,
2.1
where L is a linear operator, N is a nonlinear operator, and gt is a nonhomogeneous term.
He has modified the general Lagrange multiplier method into an iteration method which is
called correction functional as follows 1–4, 9:
un1 t un t t
μ Lun r N u
n r − gr dr,
2.2
0
where μ is a general Lagrange multiplier, which can be identified optimally via the variational
theory 21. The subscript n denotes the nth approximation, and u
n is considered as a
restricted variation 1–4, that is, δu
n 0. Employing the restricted variation in 2.2 makes
it easy to compute the Lagrange multiplier; see 22, 23. It is shown that this method is very
effective and easy and can solve a large class of nonlinear problems. For linear problems, its
exact solution can be obtained only one iteration because μ can be exactly identified.
Boundary Value Problems
3
3. Homotopy Perturbation Method
In this section, we will present a review of the homotopy perturbation method. To clarify the
basic idea of the HPM 5–8, we consider the following nonlinear differential equation:
Au − gr 0,
r ∈ Ω,
3.1
∂u
0,
B u,
∂n
r ∈ Γ,
3.2
with boundary conditions
where A is a general differential operator, B is a boundary operator, gr is a known analytic
function, and Γ is the boundary of the domain Ω. The operator A can, generally speaking, be
divided into parts L and N while N is nonlinear. Equation 3.1, therefore, can be rewritten
as follows:
Lu Nu − gr 0.
3.3
By the homotopy technique, we construct a homotopy as follows:
ν r, p : Ω × 0, 1 −→ R,
3.4
which satisfies
H ν, p 1 − p Lν − Lu0 p Aν − gr 0,
p ∈ 0, 1, r ∈ Ω,
3.5
H ν, p Lν − Lu0 pLu0 p Nν − gr 0,
p ∈ 0, 1, r ∈ Ω,
3.6
or
where p ∈ 0, 1 is an embedding parameter, and u0 is an initial approximation of 3.1 which
satisfies the boundary conditions. Obviously, from 3.5, we have
Hν, 0 Lν − Lu0 0,
Hν, 1 Aν − gr 0.
3.7
The changing process of p from zero to unity is just that of νr, p from u0 r to ur. In
topology, this is called deformation and Lν − Lu0 , and Aν − gr are called homotopic.
According to HPM, we can assume that the solution of 3.5 can be written as a power series
in p:
ν ν0 pν1 p2 ν2 · · · .
3.8
4
Boundary Value Problems
Setting p 1 results in the approximate solution 3.2:
u lim ν ν0 ν1 ν2 · · · .
p→1
3.9
The coupling of the perturbation method and the homotopy method is called the homotopy
perturbation method which has eliminated limitations of the traditional perturbation
method. On the other hand, the proposed technique can take full advantage of the traditional
perturbations techniques.
4. Applying HPM to Solve Sturm-Liouville Problem
To solve 1.1, by means of homotopy perturbation method, we choose linear operator
d
d
L yt −
pt yt ,
dt
dt
4.1
with the property Lc1 0, where c1 is constant of integration and suggests that we define
a nonlinear operator as Ny qt − λrtyt. Also gt is known analytic function
representing the nonhomogeneous term. Therefore, 1.1 can be rewritten as follows:
L y N y − g 0.
4.2
By the homotopy perturbation technique proposed by He 5–8, we can construct a homotopy
Y t, p : −l, l × 0, 1 −→ R,
d
d
d
H Y, p 1 − p
pt Y t −
pty0 t
dt
dt
dt
d
d
p
pt Y t − qt − λrt Y t gt 0,
dt
dt
4.3
d
d
d
d
d
d
H Y, p pt Y t −
pt y0 t p
pt y0 t
dt
dt
dt
dt
dt
dt
− p qt − λrt Y t − gt 0.
4.4
or
One may now try to obtain a solution of 4.2 in the form
Y t Y0 t pY1 t p2 Y2 t · · · ,
4.5
Boundary Value Problems
5
where the Yi t for i 0, 1, 2, . . . are functions yet to be determined. Substituting 4.5 into
4.4 yields
d
d
d
d
d
d
d
d
Y0 t p Y1 t p2 Y2 t · · ·
−
pt
pt y0 t p
pt y0 t
dt
dt
dt
dt
dt
dt
dt
dt
− p qt − λrt Y0 t pY1 t p2 Y2 t · · · pgt 0.
4.6
Collecting terms of the same powers of p yields
d
d
d
d
pt Y0 t −
pt y0 t 0,
dt
dt
dt
dt
d
d
d
d
pt Y1 t pt y0 t − qt − λrt Y0 t gt 0,
p1 :
dt
dt
dt
dt
d
d
pt Y2 t − qt − λrt Y1 t 0,
p2 :
dt
dt
p0 :
4.7
..
.
pn :
d
dt
d
pt
Yn t − qt − λrt Yn−1 t 0.
dt
The initial approximation Y0 t or y0 t can be freely chosen.
5. The Applications
To incorporate our discussion above, three special cases of the Sturm-Liouville equation 1.1
will be studied.
Example 5.1. Consider the Sturm-Liouville equation
−y t λyt t,
5.1
y0 t, λ A Bt,
5.2
with initial approximation
where A and B are constants. To solve 5.1 using the VIM, we have correction functional
yn1 t, λ yn t, λ t
0
μ −yn r, λ λyn r, λ − r dr,
5.3
6
Boundary Value Problems
where μ μr, t; λ is Lagrange multiplier. Making the above correction functional stationary,
we can obtain the following stationary conditions:
μ r, t; λ − λμr, t; λ 0,
1 − μ r, t; λ|rt 0,
5.4
μr, t; λ|rt 0.
The Lagrange multiplier can, therefore, be identified as
1
μr, t; λ √
λ
e
√
λr−t
− e−
2
√
λr−t
1
λr − t .
√ sinh
λ
5.5
Substituting 5.5 for correction functional 5.3, we have the following iteration formula:
yn1 t, λ yn t, λ t
sinh
0
λr − t
−yn r, λ λyn r, λ − r dr.
5.6
Using the iteration formula 5.6 and initial approximation 5.2, we get
t
1
1
y1 t, λ A cosh
sinh
λt √ B λt − .
λ
λ
λ
5.7
In the same way, we obtain
yn t, λ A cosh
t
1
1
sinh
λt √ B λt − ,
λ
λ
λ
n ≥ 2,
5.8
which means that
t
1
1
yt, λ A cosh
sinh
λt √ B λt −
λ
λ
λ
5.9
is the exact solution of 5.1.
In order to solve 5.1 using the HPM according to 4.4, we can readily construct a
homotopy which satisfies
d2
d2
d2
H Y, p 1 − p
Y t − 2 y0 t p
Y t − λY t t 0,
dt2
dt
dt2
p ∈ 0, 1,
5.10
or
d2 d2
H Y, p 2 Y t − y0 t p
y0 t − λY t t 0.
dt
dt2
5.11
Boundary Value Problems
7
We consider Y t as
Y t Y t, λ Y0 t, λ pY1 t, λ p2 Y2 t, λ · · · .
5.12
Substituting 5.12 into 5.11, collecting terms of the same power, and using initial
approximation, we have the following set of linear equations:
d2
Y0 t, λ 0,
dt2
d2
d2 p1 : 2 Y1 t, λ − 2 y0 t, λ − λY0 t, λ t 0,
dt
dt
d2
p2 : 2 Y2 t, λ − λY1 t, λ 0,
dt
..
.
p0 :
pn :
5.13
d2
Yn t, λ − λYn−1 t, λ 0.
dt2
Solving the above equations, we have
Y0 t, λ A Bt,
⎛
⎛
√ 3 ⎞
√ 2 ⎞
λt
λt
3
B ⎜
⎟
⎟ t
⎜
Y1 t, λ A⎝1 ⎠ √ ⎝ λt ⎠− ,
2!
3!
6
λ
⎛
⎜
Y2 t, λ A⎝1 √ 2
λt
2!
⎛
√ 4 ⎞
√ 3 √ 5 ⎞
λt
λt
λt
5
B ⎜
⎟
⎟ λt
,
⎠ √ ⎝ λt ⎠−
4!
3!
5!
120
λ
..
.
⎛
⎜
Yn t, λ A⎝1 √ 2
λt
2!
−1
n
⎛
√ 2n ⎞
√ 2n1 ⎞
√ 3
λt
λt
λt
B
⎜
⎟
⎟
··· ··· ⎠ √ ⎝ λt ⎠
3!
2n!
2n 1!
λ
λn1
2n 1!
.
5.14
Continuing in this manner, we can obtain
Y t, λ A cosh
t
1
1
sinh
λt √ B λt − ,
λ
λ
λ
which is exactly the same as that obtained by VIM.
5.15
8
Boundary Value Problems
Example 5.2. As another example, we consider Sturm-Liouville problem
−y t t − λyt 0,
t ≥ 0,
5.16
with initial conditions
y0 A,
y 0 B,
5.17
where A and B are constants. To solve 5.16 by means of variational method, we construct a
correction functional
yn1 t, λ yn t, λ t
0
μ −yn r, λ r yn r, λ − λyn r, λ dr,
5.18
where μ μr, t; λ is the Lagrange multiplier and yn denotes restricted variation that is
δyn 0. Then, we have
δyn1 t, λ δyn t, λ δ
t
0
μ −yn r, λ r yn r, λ − λyn r, λ dr.
5.19
Calculus of variations and integration by parts give the stationary conditions
μ r, t; λ λμr, t; λ 0,
1 μ r, t; λ|rt 0,
5.20
μr, t; λ|rt 0,
for which the Lagrange multiplier μ should satisfy. The Lagrange multiplier can, therefore,
be identified as
1
λr − t .
μr, t; λ − √ sin
λ
5.21
Substituting 5.21 into correction functional 5.18 results in the following iteration formula:
yn1 t, λ yn t, λ −
t
0
1
λr − t −yn r, λ ryn r, λ − λyn r, λ dr.
√ sin
λ
5.22
Boundary Value Problems
9
According to initial conditions 5.17, it is natural to choose initial approximation y0 t, λ A Bt. Using the above variational formula 5.22, we can obtain the following result:
⎞
⎛ √ 3 √ 5
λt
λt
A ⎜
⎟
−
···⎠
λt 3/2 ⎝
y1 t, λ A cos
3!
5!
λ
⎞
⎛ √ 4 √ 6
λt
λt
2B ⎜
B
⎟
−
· · · ⎠,
λt 2 ⎝
√ sin
4!
6!
λ
λ
5.23
..
..
In order to solve system 5.16-5.17 using HPM, after applying HPM and rearranging based
on powers of p-terms, we have
p0 :
d2
d2 Y0 t, λ − 2 y0 t, λ 0,
2
dt
dt
p1 :
d2
d2 λ
−
y0 t, λ −t λY0 t, λ 0,
Y
t,
1
2
2
dt
dt
d2
p2 : 2 Y2 t, λ −t λY1 t, λ 0,
dt
5.24
..
..
Solving the above equations, we get
⎛
⎛ √ 3 ⎞
⎛ √ 3 ⎞⎞
√ 2 ⎞
λt
λt
λt
A ⎜
B ⎜
⎟
⎟
⎟⎟
⎜
⎜
Y1 t, λ A⎝1 −
⎠ 3/2 ⎝
⎠ √ ⎝ λt − ⎝
⎠⎠
2!
3!
3!
λ
λ
⎛
⎛ √ 4 ⎞
λt
2B ⎜
⎟
2⎝
⎠,
4!
λ
⎛
⎜
Y2 t, λ A⎝1 −
√ 2
λt
⎛
2!
B ⎜
√ ⎝ λt −
λ
⎛ √ 3 √ 5 ⎞
√ 4 ⎞
λt
λt
λt
A
⎟
⎟
⎜
−
⎠ 3/2 ⎝
⎠
4!
3!
5!
λ
√ 3
λt
3!
⎛ √ 4 √ 6 ⎞
√ 5 ⎞
λt
λt
λt
⎟ 2B ⎜
⎟
−
⎠ 2⎝
⎠,
5!
4!
6!
λ
..
..
5.25
10
Boundary Value Problems
Example 5.3. Finally, we consider eigenvalue Sturm-Liouville problem
−y t − λyt 0,
t ∈ −l, l, l > 0,
5.26
yl 0.
5.27
along with the Dirichlet boundary conditions
y−l 0,
To solve 5.26 by means of variational method, we construct a correction functional for 5.26
that reads as
yn1 t, λ yn t, λ t
0
μ −yn r, λ − λyn r, λ dr,
5.28
where μ μr, t; λ is Lagrange multiplier. Following the discussion presented in the previous
example, we obtain the following iteration formula:
yn1 t, λ yn t, λ −
t
0
1
λr − t −yn r, λ − λyn r, λ dr.
√ sin
λ
5.29
Let us begin with an initial approximation y0 t, λ A Bt, where A and B are constants to
be determined. Substituting the proposed initial iterate y0 t, λ in 5.29 gives
y1 t, λ A cos
B
λt √ sin
λt .
λ
5.30
In the same way, we obtain
yn t, λ A cos
B
λt √ sin
λt ,
λ
n ≥ 2.
5.31
So, we can derive that
yt, λ A cos
is the exact solution of 5.26.
B
λt √ sin
λt
λ
5.32
Boundary Value Problems
11
In order to solve 5.26 using HPM, similar to previous examples, after applying HPM
and rearranging based on powers of p-terms, we have
p0 :
d2
d2 λ
−
y0 t, λ 0,
Y
t,
0
dt2
dt2
p1 :
d2
d2 Y1 t, λ − 2 y0 t, λ λY0 t, λ 0,
2
dt
dt
p2 :
d2
Y2 t, λ λY1 t, λ 0,
dt2
5.33
..
.
pn :
d
Yn t, λ λYn−1 t, λ 0.
dt
Now, we choose y0 t, λ A Bt. Solving the above sets of equations yields
Y0 t, λ A Bt,
⎛
⎛
√ 3 ⎞
√ 2 ⎞
λt
λt
B ⎜
⎟
⎟
⎜
Y1 t, λ A⎝1 −
⎠ √ ⎝ λt −
⎠,
2!
3!
λ
⎛
⎜
Y2 t, λ A⎝1 −
√ 2
λt
2!
⎛
√ 4 ⎞
√ 3 √ 5 ⎞
λt
λt
λt
B ⎜
⎟
⎟
⎠ √ ⎝ λt −
⎠,
4!
3!
5!
λ
5.34
..
.
⎛
⎜
Yn t, λ A⎝1 −
√ 2
λt
⎛
2!
B ⎜
√ ⎝ λt −
λ
√ 4
λt
4!
√ 3
λt
3!
√ 2n ⎞
λt
⎟
· · · −1n
⎠
2n!
√ 5
λt
5!
√ 2n1 ⎞
λt
⎟
· · · −1n
⎠.
2n 1!
Hence, from 4.4 we get
Y t, λ A cos
B
λt √ sin
λt ,
λ
5.35
12
Boundary Value Problems
which is exactly the same as that obtained by VIM. Now, we use the boundary condition
5.27 to obtain eigenvalue and eigenfunctions of 5.26. Imposing the boundary conditions
in 5.35 yields
B
λl − √ sin
λl 0,
λ
B
Y l, λ A cos
λl √ sin
λl 0.
λ
Y −l, λ A cos
5.36
So, there are two infinite sequences of eigenvalues λm :
2m − 1π
λm 2l
mπ 2
,
λm l
2
m 1, 2, . . . ,
,
5.37
m 1, 2, . . . .
Thus, corresponding linearly nontrivial solutions are
2m − 1π
t , m 1, 2, . . . ,
2l
mπ l
vm t B
sin
t , m 1, 2, . . . .
mπ
l
um t A cos
5.38
Since um t and vm t are of class CI, R, that is, are continuous real-valued functions of
t ∈ −l, l, using the definition of inner product on CI, R, that is,
um , vm l
−l
um xvm xdx,
um , vm ∈ CI, R,
5.39
and the norm induced by inner product
um 2 2 um , um l
−l
um xvm xdx,
5.40
we get the normalization constants as
1
A √ ,
l
mπ
B √ .
l3
5.41
Boundary Value Problems
13
Consequently, we obtain
1
2m − 1π
t , m 1, 2, . . . ,
u
m t √ cos
2l
l
mπ 1
vm t √ sin
t , m 1, 2, . . . ,
l
l
5.42
where u
m t and vm t are normalized eigenfunctions, that is, u
m t um t/
um 2 and
vm t vm t/
vm 2 .
6. Conclusion
In this work, we proposed variational method and compared with homotopy perturbation
method to solve ordinary Sturm-Liouville differential equation. The variational iteration
algorithm used in this paper is the variational iteration algorithm-I; there are also variational
iteration algorithm-II and variational iteration algorithm-III 24, which can also be used for
the present paper. It may be concluded that the two methods are powerful and efficient
techniques to find exact as well as approximate solutions for wide classes of ordinary
differential equations.
References
1 J.-H. He, “A new approach to nonlinear partial differential equations,” Communications in Nonlinear
Science and Numerical Simulation, vol. 2, no. 4, pp. 230–235, 1997.
2 J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous
media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.
3 J.-H. He, “Approximate solution of nonlinear differential equations with convolution product
nonlinearities,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 69–73,
1998.
4 J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,”
International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.
5 J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering,
vol. 178, no. 3-4, pp. 257–262, 1999.
6 J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied
Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003.
7 J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear
problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000.
8 J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A,
vol. 350, no. 1-2, pp. 87–88, 2006.
9 J.-H. He, “Variational iteration method for autonomous ordinary differential systems,” Applied
Mathematics and Computation, vol. 114, no. 2-3, pp. 115–123, 2000.
10 J.-H. He, “Variational iteration method for delay differential equations,” Communications in Nonlinear
Science and Numerical Simulation, vol. 2, no. 4, pp. 235–236, 1997.
11 M. A. Abdou and A. A. Soliman, “New applications of variational iteration method,” Physica D, vol.
211, no. 1-2, pp. 1–8, 2005.
12 M. A. Abdou and A. A. Soliman, “Variational iteration method for solving Burger’s and coupled
Burger’s equations,” Journal of Computational and Applied Mathematics, vol. 181, no. 2, pp. 245–251,
2005.
13 S. Momani and S. Abuasad, “Application of He’s variational iteration method to Helmholtz
equation,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119–1123, 2006.
14
Boundary Value Problems
14 M. El-Shahed, “Application of He’s homotopy perturbation method to Volterra’s integro-differential
equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 163–168,
2005.
15 J.-H. He, “Application of He’s homotopy perturbation method to nonlinear wave equations,” Chaos
Solitons & Fractals, vol. 167, no. 1-2, pp. 69–73, 1998.
16 S. Abbasbandy, “Homotopy perturbation method for quadratic Riccati differential equation and
comparison with Adomian’s decomposition method,” Applied Mathematics and Computation, vol. 172,
no. 1, pp. 485–490, 2006.
17 D. D. Ganji and A. Sadighi, “Application of He’s homotopy-perturbation method to nonlinear
coupled systems of reaction-diffusion equations,” International Journal of Nonlinear Sciences and
Numerical Simulation, vol. 7, no. 4, pp. 411–418, 2006.
18 J.-H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” International
Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 207–208, 2005.
19 J.-H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,” Applied
Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004.
20 M. Rafei and D. D. Ganji, “Explicit solutions of Helmholtz equation and fifth-order KdV equation
using homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical
Simulation, vol. 7, no. 3, pp. 321–328, 2006.
21 J.-H. He, “Variational principles for some nonlinear partial differential equations with variable
coefficients,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 847–851, 2004.
22 J.-H. He, “Semi-inverse method of establishing generalized variational principles for fluid mechanics
with emphasis on turbomachinery aerodynamics,” International Journal of Turbo and Jet Engines, vol.
14, no. 1, pp. 23–28, 1997.
23 J.-H. He, Y.-Q. Wan, and Q. Guo, “An iteration formulation for normalized diode characteristics,”
International Journal of Circuit Theory and Applications, vol. 32, no. 6, pp. 629–632, 2004.
24 J.-H. He, G.-C. Wu, and F. Austin, “The variational iteration method which should be followed,”
Nonlinear Science Letters A, vol. 1, no. 1, pp. 1–30, 2010.